Expansions in Noninteger Bases.
Vilmos Komornik · DOAJ (DOAJ: Directory of Open Access Journals) · 2011
In [2] Erdös, Horváth and Joó found a curious uniqueness property for the expansions of the number 1 in some noninteger bases q. These numbers q were then characterized in [3]. Using that characterization the smallest such number was determined in [8]. Allouche and Cosnard [1] discovered that this number is closely related to the classical Thue-Morse sequence, and using this relation they proved the transcendence of this number. Erdös and Joó [4] also constructed, for each positive integer N, bases q for which the number 1 has exactly N different expansions. Their construction was generalized in [5]. The characterization of these numbers remains an open question for N > 1. The purpose of this paper is to give a useful sufficient condition for N = 2 and to use this for the construction of small numbers q for which the number 1 has exactly 2 different expansions.